\subsection*{2012-06-27}

\begin{itemize}
\item  
  Coordinates for AdS -- the Poincare only covers a patch -- aka the geodesics are incomplete, we need to either add boundary or more patches. 
  Note: it is good we are in AdS, b/c otherwise adding boundary is messed up (GR doesn't like boundaries). We can get a global AdS with different coordinates.


  If we solve EFE in arbitrary dimension, see Allan's mathematica. 
  Tons of tricks involving gauge invariance. 
  We get an integration constant -- this corresponds to the mass of an object. 
  Note that if we set $\Lambda = 0$, we recover Schwarzschild metric. 
  Aha. 
  Also the metric goes like $1 + c_1/r + \Lambda r^2$ -- the $r^2$ term is due to the AdS curvature. 
  If radius of curvature is huge then this is like Schwarzschild because we can't tell we are in flat.

  If you do the ADM mass calculation we will find that indeed c1 is the mass up to a factor of 2 (Exercise)

  Set up a system in 3+1 dimensions with no time-reversal invariance and only SO(2) not SO(3) symmetry, in other words only one axis of rotational invariance. 
  We will get what's called a Kerr-AdS black hole. 
  This has another d.o.f. 
  Solving is messy, but if feeling punchy go for it.

\item
  The Problem we are working on.

  Suppose you throw goats into a BH. 
  The BH (in AdS or whatever) is originally some metric $g_0$, but now we have an out of equilibrium perturbation h. 
  So $g = g_0 + h$. 
  This $h$ will dissipate over time and eventually relax into a BH with slightly bigger event horizon. 
  We can solve EFE's to get a diff eq for h (series expansion or whatnot).

  This is a very messy thing to do. 
  Allan and his postdoc have written code that does this.

  However, it is a fact of gravity that, in the limit near the event horizon, in low energy, GR reduces to a hydrodynamics problem. 
  Navier-Stokes plus hydrodynamic terms (e.g. viscosity, etc) $h_{ti} \sim v_i$

  In fact, there is a "duality" between solutions of ANY hydrodynamic problem, and a solution to the EFE. 
  ANY hydrodynamics problem means inhomogenieities, etc, and it will be dual to an inhomogenous metric, with unicorns that have nasty sharp pointy teeth.

  Hydrodynamics is much simpler to solve numerically than GR! GR is d+1 dimensions; hydrodynamics is d dimensions; GR has all these degrees of freedom with diffeomorphism invariance from gauge symmetry; NS does not.

  Our job is to solve the hydrodynamics problem.

  Why is this interesting? Because the GR problem is, by holography, dual to a quantum liquid that is super cool and hard. So $\mathrm{QFT}_d = \mathrm{GR}_{d+1}$. 
  Near the horizon, $\mathrm{GR}_{d+1}$ is described by a liquid in $d$. 
  Thus $\mathrm{QFT}_d$ is described by a liquid in $d$. 
  Unsurprising but the trick is --what liquid?

\end{itemize}


  So our job is:

\begin{enumerate}
\item  Papers (reference). Allan will email us a bunch of references. Some of them

\begin{itemize}
 \item  Set up the fluid/gravity "duality"

 \item Get $g_{\mu}$ nu in terms of $v(x,t)$. 
 One paper does it in 3+1, one does it in 4+1.
\end{itemize}

\item Compute (in some coords) the e.o.m. for $v$ and $\pi$ (canonical momentum) in 3+1 gravity (2+1 liquid).

\item Write code to solve for $v(x,t)$

\item Plug it in to get the metric $g(v)$
\end{enumerate}
